Printing Pictures on the Surface of Polyhedra

Building a dodecahedron with the cube edges marked

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When Gareth asked if I could make a dodecahedron with the five inscribed cubes marked on its surface, I wasn't immediately sure how best to do it. All my other polyhedral pictures are basically spherical images, without any specific reference to the polyhedron they'll end up drawn on; you can re-project them on to a different solid relatively trivially. But to construct the net of a dodecahedron with an extra set of lines drawn between its vertices, this seemed like overkill: what I'd be drawing would be fundamentally based on the vertices I already knew about, and there would be no need to project it on to a different solid at all.

Nonetheless, it turned out, using the existing spherical-image mechanism was still the easiest way to go about it, not least because in order to do it any other way I'd have had to modify my net-drawing utility again, and it's complicated enough that that would have outweighed any saving due to conceptual parsimony.

And, in fact, it wasn't too hard to do it directly using the spherical image code. First I found the polyhedron description file I was going to use for my dodecahedron; that has the 3-D coordinates of the 20 vertices already specified. So it was just a question of identifying the eight subsets of those vertices which formed cubes, and directly writing a PostScript spherical-image description consisting of appropriately coloured straight lines between those points in space. My projection code doesn't change the orientation of the image relative to the polyhedron, so this naturally produced the right result when projected on to the same dodecahedron description file I'd started with.

Of course, geodesics in spherical image space (great circle arcs) project into straight lines on the resulting solid, so there wasn't even any difficulty in making the lines straight.

Identifying the actual cubes was a moderately frustrating ten-minute task with a pencil and paper, in which I figured out the quick and easy way to do it just as I was finishing the last cube. But it only had to be done once, and then it was just a matter of tweaking the colours and line thicknesses.


Here's the PDF net of the cube-marked dodecahedron.

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(last modified on Sun May 7 14:33:22 2017)